## Foundations of Set TheoryFoundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. |

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### Contents

1 | |

15 | |

CHAPTER III TYPETHEORETICAL APPROACHES | 154 |

CHAPTER IV INTUITIONISTIC CONCEPTIONS OF MATHEMATICS | 210 |

CHAPTER V METAMATHEMATICAL AND SEMANTICAL APPROACHES | 275 |

Bibliography | 346 |

391 | |

397 | |

399 | |

### Common terms and phrases

algebra Amsterdam antinomies arguments arithmetic attitude axiom of choice axiom of comprehension axiom of foundation axiom of infinity axiom of pairing axiom schema Axiom VIII axiomatic Bernays Brouwer Cantor's cardinal numbers Carnap Chapter classical concept condition 3(x consistency proofs construction continuum hypothesis contradiction defined definition denote denumerable equivalent existence extensionality finite number first-order first-order predicate calculus formal system formula function given Gödel hence Heyting Hilbert implies impredicative inaccessible cardinal induction infinite set integers interpretation intuitionism intuitionistic intuitive Kleene language layer logic Math mathematicians mathematics means mentioned metamathematical method Mostowski natural numbers Neumann notion of set null-set number theory objects obtained ordinal power-set predicate calculus principle problem provable prove quantifiers Quine real numbers recursive relation semantical sentences sequence set theory statement symbols system of set Tarski theorem tion type theory valid variables well-founded well-ordering well-ordering theorem Zermelo